NEUTROSOPHIC SOFT MULTI-ATTRIBUTE DECISION MAKING BASED ON GREY RELATIONAL PROJECTION METHOD by Mia Amalia

Neutrosophic Sets and Systems, Vol. 11, 2016

University of New Mexico

NEUTROSOPHIC SOFT MULTI-ATTRIBUTE DECISION MAKING BASED ON GREY RELATIONAL PROJECTION METHOD Partha Pratim Dey1, Surapati Pramanik2, and Bibhas C. Giri3 1

Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India. E-mail: parsur.fuzz@gmail.com Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District â&#x20AC;&#x201C;North 24 Parganas, Pin code-743126,West Bengal, India. E-mail: sura_pati@yahoo.co.in 3 Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India. E-mail: bcgiri.jumath@gmail.com

Abstract. The present paper proposes neutrosophic soft multi-attribute decision making based on grey relational projection method. Neutrosophic soft sets is a combination of neutrosophic sets and soft sets and it is a new mathematical apparatus to deal with realistic problems in the fields of medical sciences, economics, engineering, etc. The rating of alternatives with respect to choice parameters is represented in terms of neutrosophic soft sets.

The weights of the choice parameters are completely unknown to the decision maker and information entropy method is used to determine unknown weights. Then, grey relational projection method is applied in order to obtain the ranking order of all alternatives. Finally, an illustrative numerical example is solved to demonstrate the practicality and effectiveness of the proposed approach.

Keywords: Neutrosophic sets; Neutrosophic soft sets; Grey relational projection method; Multi-attribute decision making.

1 Introduction In real life, we often encounter many multi-attribute decision making (MADM) problems that cannot be described in terms of crisp numbers due to inderminacy and inconsistency of the problems. Zadeh [1] incorporated the degree of membership and proposed the notion of fuzzy set to handle uncertainty. Atanassov [2] introduced the degree of non-membership and defined intuitionistic fuzzy set to deal with imprecise or uncertain decision information. Smarandache [3, 4, 5, 6] initiated the idea of neutrosophic sets (NSs) by using the degree of indeterminacy as independent component to deal with problems involving imprecise, indeterminate and inconsistent information which usually exist in real situations. In NSs, indeterminacy is quantified and the truth-membership, indeterminacy-membership, falsitymembership functions are independent and they assume the value from ] -0, 1+ [. However, from scientific and realistic point of view Wang et al. [7] proposed single valued NSs (SVNSs) and then presented the set theoretic operators and various properties of SVNSs. Molodtsov [8] introduced the soft set theory for dealing with uncertain, fuzzy, not clearly described objects in 1999. Maji et al. [9] applied the soft set theory for solving decision making problem. Maji et al. [10] also

defined the operations AND, OR, union, intersection of two soft sets and also proved several propositions on soft set operations. However, Ali et al. [11] and Yang [12] pointed out that some assertions of Maji et al. [10] are not true in general, by counterexamples. The soft set theory have received a great deal of attention from the researchers and many researchers have combined soft sets with other sets to make different hybrid structures like fuzzy soft sets [13], intuitionistic fuzzy soft sets [14], vague soft sets [15] generalized fuzzy soft sets [16], generalized intuitionistic fuzzy soft [17], possibility vague soft set [18], etc. The different hybrid systems have had quite impact on solving different practical decision making problems such as medical diagnosis [16, 18], plot selection, object recognition [19], etc where data set are imprecise and uncertain. Maji et al. [13, 14] incorporated fuzzy soft sets and intuitionistic soft sets based on the nature of the parameters involved in the soft sets. CaÄ&#x;man et al. [20] redefined fuzzy soft sets and their properties and then developed fuzzy soft aggregation operator for decision problems. Recently, Maji [21] introduced the concept of neutrosophic soft sets (NSSs) which is a combination of neutrosophic sets [3, 4, 5, 6] and soft sets [8], where the parameters are neutrosophic sets. He also introduced several definitions and operations on NSSs and presented an application of NSSs in house selection problem. Maji

Partha Pratim Dey, Surapati Pramanik, and Bibhas C. Giri, Neutrosophic soft multi-attribute decision making based on grey relational projection method

Neutrosophic Sets and Systems, Vol.11, 2016

[22] further studied weighted NSSs by imposing some weights on the parameters. Based on the concept of weighted NSSs, Maji [23] solved a multi-criteria decision making problem. MADM problem generally comprises of selecting the most suitable alternative from a set of alternatives with respect to their attributes and it has received much attention to the researchers in the field of decision science, management, economics, investment [24, 25], school choice [26], etc. Grey relational analysis (GRA) [27] is an effective tool for modeling MADM problems with complicated interrelationships between numerous factors and variables. GRA is applied in a range of MADM problems such as agriculture, economics, hiring distribution [28], marketing, power distribution systems [29], personal selection, teacher selection [30], etc. Biswas et al. [24] investigated entropy based GRA method for solving MADM problems under single valued neutrosophic assessments. Biswas et al. [25] also studied GRA based single valued neutrosophic MADM problems with incomplete weight information. Mondal and Pramanik [26] presented a methodological approach to select the best elementary school for children using neutrosophic MADM with interval weight information based on GRA. Mondal and Pramanik [31] also developed rough neutrosophic MADM based on modified GRA. Zhang et al.. [32] developed a new grey relational projection (GRP) method for solving MADM problems in which the attribute value takes the form of intuitionistic trapezoidal fuzzy number, and the attribute weights are unknown. In this paper, we have extended the concept of Zhang et al. [32] to develop a methodology for solving neutrosophic soft MADM problems based on grey relational projection method with unknown weight information. Rest of the paper is organized as follows. Section 2 presents some definitions concerning NS, SVNS, soft sets, and neutrosophic soft sets. A neutrosophic soft MADM based on GRP method is discussed in Section 3. In Section 4, we have solved a numerical example in order to demonstrate the proposed procedure. Finally, Section 5 concludes the paper.

A= {x, TA ( x), I A ( x), FA ( x)  x X}. where, TA ( x) , I A ( x ) , FA ( x) : X  ]-0, 1+[ are the truthmembership,

indeterminacy-membership,

In this section we briefly present some basic definitions regarding NSs, SVNSs, soft sets, and NSSs. 2.1 Neutrosophic set Definition 1 [3, 4, 5, 6] Consider X be a universal space of objects (points) with generic element in X denoted by x.

falsity-

membership functions, respectively and -0  sup TA ( x) + sup I A ( x ) + sup FA ( x)  3+. We consider the NS which assmes the value from the subset of [0, 1] because] -0, 1+ [ will be hard to apply in real world science and engineering problems. Definition 2 [7] Let X be a universal space of points with ~ generic element in X represented by x. Then a SVNS N

 X is characterized by a truth-membership function TN~ ( x) , a indeterminacy-membership function I N~ ( x) , and

a falsity-membership function FN~ ( x) with TN~ ( x) , I N~ ( x) , FN~ ( x) : X  [0, 1] for each point x X and we have,

0  sup TN~ ( x) + sup I N~ ( x) + sup FN~ ( x)  3. ~ Definition 3 [7] The complement of a SVNS N is ~ represented by N C and is defined by

TN~ C ( x) = FN~ ( x) ; I N~ C ( x) = 1 - I N~ ( x) ; FN~C ( x) = TN~ ( x) ~ ~ Definition 4 [7] For two SVNSs N A and N B ~ N A = {x, TN~ ( x), I N~ ( x), FN~ ( x) A

 x X}

and ~ N B = {x, TN~ ( x), I N~ ( x), FN~ ( x) B

 x X}

~ ~ N A  N B if and only if

TN~ ( x)  TN~ ( x) ; I N~ ( x)  I N~ ( x) ; FN~ ( x)  FN~ ( x) B

2 Preliminaries

and

~ ~ 2. N A = N B if and only if

TN~ ( x) = TN~ ( x); I N~ ( x) = I N~ ( x) ; FN~ ( x) = FN~ ( x);  x  A

X. ~ ~ 3. N A  N B

Then a NS is defined as follows:

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x, max{T N~ ( x), TN~ ( x)}, min{I A

~ NA

( x), I N~ ( x)}, B

min{ FN~ ( x), FN~ ( x)} | x  X A

objects and F = {f1, f2, f3, f4} be a set of parameters. Here,

}

f1, f2, f3, f4 stand for the parameters ‘very large’, ‘large’,

‘attractive’, ‘expensive’ respectively. Suppose that,

~ ~ 4. N A  N B

M (very large) = {< u1, 0.8, 0.3, 0.4>, < u2, 0.7, 0.3, 0.5>,

x, min{T N~ ( x), TN~ ( x)}, max{I N~ ( x), I N~ ( x)}, A

max{ FN~ ( x), FN~ ( x)} | x  X A

< u3, 0.8, 0.2, 0.3>, < u4, 0.6, 0.4, 0.5>, < u5, 0.9, 0.3, 0.3>,

}

< u6, 0.8, 0.4, 0.5>},

~ Definition 5 [7] The Hamming distance between N A =

{xi,



TN~ ( xi ), I N~ ( xi ), FN~ ( xi ) A

{xi, TN~ ( xi ), I N~ ( xi ), FN~ ( xi ) B

xiX}

~ and N B

 xi X} is defined as given

M (large) = {< u1, 0.7, 0.3, 0.2>, < u2, 0.6, 0.3, 0.4>, < u3, 0.6, 0.4, 0.4>, < u4, 0.6, 0.3, 0.2>, < u5, 0.7, 0.5, 0.4>, < u6, 0.6, 0.5, 0.6>}, M (attractive) = {< u1, 0.9, 0.2, 0.2>, < u2, 0.8, 0.3, 0.2>, < u3, 0.8, 0.2, 0.3>, < u4, 0.9, 0.4, 0.2>, < u5, 0.8, 0.5, 0.4>, <

below.

u6, 0.7, 0.4, 0.6>},

~ ~ 1 n H ( N A , N B ) =  | TN~ ( x i ) - TN~ ( x i ) | + | I N~ ( x i ) A A B 3 i 1

M (expensive) = {< u1, 0.8, 0.2, 0.3>, < u2, 0.9, 0.1, 0.2>,

I N~ ( xi ) | + | FN~ ( xi ) - FN~ ( xi ) | A B

< u6, 0.8, 0.2, 0.5>}

(1)

~ ~ with the property: 0  H ( N A , N B )  1.

< u3, 0.8, 0.3, 0.5>, < u4, 0.9, 0.3, 0.3>, < u5, 0.8, 0.4, 0.5>, Therefore, M (very large) means very large objects, M (attractive) means attractive objects, etc. Now we can

2.2 Soft sets and Neutrosophic soft sets Definition 6 [8] Suppose U is a universal set, F is a set of

represent the above NSS (M, A) over U in the form of a table (See the Table 1).

parameters and P (U) is a power set of U. Consider a nonempty set A, where A  F. A pair (M, A) is called a soft

Table 1. Tabular form of the NSSs (M, A)

set over U, where M is a mapping given by M: A P (U). Definition 7 [21] Let U be an initial universal set. Let F be a set of parameters and A be a non-empty set such that A

 F. P(U) represents the set of all neutrosophic subsets of

u1 u2 u3

U. A pair (M, A) is called a NSS over U, where M is a mapping given by M: A P (U). In other words, (M, A) over U is a parameterized family f of all neutrosophic sets over U.

u4 u5 u6

Example: Let U be the universal set of objects or points. F

f1 = very large (0.8, 0.3, 0.4) (0.7, 0.3, 0.5) (0.8, 0.2, 0.3) (0.6, 0.4, 0.5) (0.9, 0.3, 0.3) (0.8, 0.4, 0.5)

f2 = large (0.7, 0.3, 0.2) (0.6, 0.3, 0.4) (0.6, 0.4, 0.4) (0.6, 0.3, 0.2) (0.7, 0.5, 0.4) (0.6, 0.5, 0.6)

f3 = attractive (0.9, 0.2, 0.2) (0.8, 0.3, 0.2) (0.8, 0.2, 0.3) (0.9, 0.4, 0.2) (0.8, 0.5, 0.4) (0.7, 0.4, 0.6)

f4 = expensive (0.8, 0.2, 0.3) (0.9, 0.1, 0.2) (0.8, 0.3, 0.5) (0.9, 0.3, 0.3) (0.8, 0.4, 0.5) (0.8, 0.2, 0.5)

= {very large, large, medium large, medium low, low, very low, attractive, cheap, expensive} is the set of parameters and each parameter is a neutrosophic word or sentence concerning neutrosophic word. To define neutrosophic soft set means to find out very large objects, large objects, medium large objects, attractive objects, and so on. Let U

Definition 8 [21]: Consider two NSSs (M1, A) and (M2, B) over a common universe U. (M1, A) is said to be neutrosophic soft subset of (M2, B) if M1  M2, and TM1 (f) (x)  TM 2 (f) (x), I M1 (f) (x)  I M 2 (f) (x), FM1 (f) (x) 

FM 2 (f) (x),  f  A, x  U. We represent it by (M1,

A)  (M2, B).

= (u1, u2, u3, u4, u5, u6) be the universal set consisting of six Partha Pratim Dey, Surapati Pramanik, and Bibhas C. Giri, Neutrosophic soft multi-attribute decision making based on grey relational projection method

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A  B

Definition 9 [21]: Let (M1, A) and (M2, B) be two NSSs

where

over a common universe U. They are said to be equal i.e.

indeterminacy-membership

and

the

and

truth-membership, falsity-membership

(M1, A) = (M2, B) if (M1, A)  (M2, B) and (M2, B)  (M1, functions of (N, D) are as follows:

TN(f) (m) = min ( TM1 (f) (m), TM 2 (f) (m)); I N(f) (m)

A). Definition 10 [21]: Consider F = {f1, f2, …, fq} be a set of

I M1 (f) (m)  I M 2 (f) (m)

parameters. Then, the NOT of F is defined by NOT F =

{not f1, not f2, …, not fq}, where it is to be noted that NOT

FM2 (f) (m)).

and not are different operators. Definition 11 [21]: The complement of a neutrosophic soft set (M, A) is denoted by (M, A)C and is represented as (M, C

A) = (M , NOT A) with TM = I M(f) (x); FM

(f)

(x) = FM(f) (x); I M

(f)

(x)

(x) = TM(f) (x), where MC: NOT A P (U).

Definition 12 [21]: A NSS (M, A) over a universe U is called a null NSS with respect to the parameter A if

TM(f) (m) = I M(f) (m) = FM(f) (m) = 0,  f

 A,  m  U.

Definition 13 [21]: Let (M1, A) and (M2, B) be two NSSs over a common universe U. The union (M1, A) and (M2, B) is defined by (M1, A)

 (M2, B) = (M, C), where C = A

 B and the truth-membership, indeterminacymembership and falsity-membership functions are defined as follows:

TM(f) (m) = TM1 (f) (m), if f  M1 - M2,

= TM 2 (f) (m), if f  M2 – M1,

= max ( TM1 (f) (m), TM 2 (f) (m)), if f  M1  M2.

I M(f) (m) = I M1 (f) (m), if f  M1 - M2,

= I M 2 (f) (m), if f  M2 – M1, I M 1 (f) (m)  I M 2 (f) ( m)

if f  M1  M2. 2 FM(f) (m) = FM1 (f) (m), if f  M1 - M2, =

= FM 2 (f) (m), if f  M2 – M1,

= min ( FM1 (f) (m), FM2 (f) (m)), if f  M1  M2.

Definition 14 [21]: Suppose (M1, A) and (M2, B) are two NSSs over a common universe U. The intersection (M1, A) and (M2, B) is defined by (M1, A)  (M2, B) = (N, D),

; FN(f) (m) = max ( FM1 (f) (m),

3 A neutrosophic soft MADM based on grey relational projection method Assume G = {g1, g2, …, gp}, (p  2) be a discrete set of alternatives and A ={a1, a2, …, aq}, (q  2) be a set of choice parameters under consideration in a MADM problem. The rating of performance value of alternative gi, i = 1, 2, …, p with respect to the choice parameter aj, j = 1, 2, …, q is represented by a tuple tij = ( TM(a ) (gi), I M(a ) (gi), FM(a ) (gi)), where for a fixed i the value tij (i = 1, 2, …, p; j = 1, 2, …, q) denotes NSS of all the p objects. Let w = {w1, w2, …, wq} be the weight vector assigned for the choice q parameters, where 0  wj  1with  wj = 1, but specific j1 value of wj is unknown. Now the steps of decision making based on neutrosophic soft information are described as given below. j

Step 1. Construction of criterion matrix with SVNSs GRA method is appropriate for dealing with quantitative attributes. However, in the case of qualitative attribute, the performance values are taken as SVNSs. The performance values t N~ (i = 1, 2, …, p; j = 1, 2, …, q) could be arranged in the matrix called criterion matrix and whose rows are labeled by the alternatives and columns are labeled by the choice parameters. The criterion matrix is presented as follows: t11 t12 ... t1q    t 21 t 22 ... t 2q  . ... .  = . D N~ = t N~   p q . ... .  .   t p1 t p2 ... t pq  where tij = (Tij, Iij, Fij) where Tij, Iij, Fij  [0, 1] and 0  Tij + ij

Iij + Fij  3, i = 1, 2, …, p; j = 1, 2, …, q. Step 2. Determination of weights of the attributes In the decision making situation, the decision maker encounters problem of identifying the unknown attributes

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weights, where it may happen that the weights of attributes are different. In this paper, we use information entropy method in order to obtain unknown attribute weight. The entropy measure can be used when weights of attributes are dissimilar and completely unknown to the decision ~ maker. The entropy measure [33] of a SVNS N = {x, TN~ ( x), I N~ ( x), FN~ ( x) is defined as given below. ~ 1 p Ei( N ) = 1 -  (TN~ ( xi )  FN~ ( xi )) I N~ ( xi )  I CN~ ( xi ) (2) n i 1 which has the following properties: ~ ~ (i). Ei ( N ) = 0 if N is a crisp set and I N~ ( x i ) = 0,  x  X.

and ideal neutrosophic estimates un-reliability solution (INEURS). An INERS PN~ = [ p N~ , p N~ , …, p N~ ] is a 1

solution in which every element p

= <0.5, 0.5,

 j

= < T , I , F j >,

where T j = max {Tij}, I j = min {Iij}, F j = min {Fij} in the i

criteria matrix D N~ = < Tij, Iij, Fij > p  q for i = 1, 2, …, p; j = 1, 2, …, q. Also, an INEURS PN~ = [ p N~ , p N~ , …, p N~ ] is 1

a solution in which every element p

 ~ Nj

 j

= <T ,I , F > pq ,

where T j = min {Tij}, I j = max {Iij}, Fj = max {Fij} in the i

~ (ii). Ei ( N ) = 0 if TN~ ( xi ), I N~ ( xi ), FN~ ( xi )

 ~ Nj

criterion matrix D N~ = < Tij, Iij, Fij > p  q for i = 1, 2, …, p; j = 1, 2, …, q.

0.5>,  x  X. Step 4. Grey relational projection method ~ ~ ~ ~ (iii). Ei ( N1 )  Ei ( N 2 ) if N 1 is more uncertain than N 2 i.e.

3.1 Projection method Definition 15 [37, 38]: Consider a = (a1, a2, …, aq) and b

TN~ ( x i )



FN~ ( x i )

and I N~ ( x i )  I CN~ ( x i ) 1

TN~ ( xi )

FN~ ( x i ) 2

angle between vectors a and b is defined as follows:

 I N~ ( xi )  I CN~ ( xi ) . 2

= (b1, b2, …, bq) are two vectors, then cosine of included

q  j1

(a j b j )

Cos (a, b) =

~ ~ (iv). Ei ( N ) = Ei ( N C ),  x  X.

q 2  j j1

a 

Therefore, the entropy value Ej of the j-th attribute can be

(j = 1, 2,…, q).

Definition 16 [37, 38]: Let a = (a1, a2, …, aq) be a vector, (3)

Here, 0  Ej  1 and according to Hwang and Yoon [34] and Wang and Zhang [35] the entropy weight of the j-th attribute is defined as follows:

1 Ej 1 Ej

, with 0  wj

Obviously, 0 < Cos (a, b)  1, and the direction of a and b

b).

1 p Ej = 1 -  (Tij (x i )  Fij (x i )) I ij (x i )  I ijC (x i ) , q i 1

q  j1

is more accordant according to the bigger value of Cos (a,

obtained as follows:

wj =

(5) q 2  j j1

 1 and

q  j1

wj =1

(4)

then norm of a is given by || a || =

q 2  j j1

(6)

The direction and norm are two important parts of a vector. However, Cos (a, b) can only compute whether their directions are accordant, but cannot determine the magnitude of norm. Therefore, the closeness degree of two vectors can be defined by the projection value in order to take the norm magnitude and cosine of included angle together.

Step 3. Determination of ideal neutrosophic estimates

Definition 17 [37, 38]: Let a = (a1, a2, …, aq) and b = (b1,

reliability solution (INERS) and ideal neutrosophic

b2, …, bq) be two vectors, then the projection of vector a

estimates un-reliability solution (INEURS)

onto vector b is defined as follows:

Dezart [36] proposed the idea of single valued neutrosophic cube. From this cube one can easily obtain ideal neutrosophic estimates reliability solution (INERS) Partha Pratim Dey, Surapati Pramanik, and Bibhas C. Giri, Neutrosophic soft multi-attribute decision making based on grey relational projection method

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 0 = (  01 ,  02 , …,  0q ) = (1, 1, …, 1)

Pr (a) = || a || Cos (a, b) = q 2  j j1

q  j1

(a j b j )



(a j b j )

q 2  j j1

q  j1

q 2  j j1

a 

(7) q 2  j j1

b to the vector a is.

3.2 Grey correlation projection method The grey correlation projection method is a combination of grey correlation method and projection method. The method is presented in the following steps. Step-1. The grey relational coefficient of each alternative from INERS is obtained from the following formula:

 ij =

min min   σ max max  i

 ij

(8)

  σ max max  ij i

INEURS is:

 0 = (  01 ,  02 , …,  0q ) = (1, 1, …, 1) Step-3. Weighted neutrosophic grey correlation coefficient matrix G between every alternative and INERS is

formulated as given below.

between t N~ and p N~ , (i = 1, 2, ..., p; j = 1, 2, ..., q).

 w1 11    w1 21  G+ = . .    w1 p1  The weighted

Also, the grey relational coefficient of each alternative from INEURS is obtained from the formula given below:

 =  ij

min min  ij  σ max max  ij i

 ij

(9)

  σ max max  ij i

between every

 11  12 ...  1q       21  12 ...  2q      = . . ... .  . . ... .      p1  12 ...  pq    Similarly, the correlation coefficient between INEURS and

where  ij = d( t N~ , p N~ ) = Hamming distance j



alternative and INEURS is constructed as follows.

The bigger the value of Pr (a) is, the more close the vector

 ij

Grey correlation coefficient matrix

wq  1q    w2  12 ... wq  2q   . ... .   . ... .    w2  12 ... wq  pq  correlation coefficient between INERS and w2  12

...

where  ij = d( t N~ , p N~ ) = Hamming distance

INERS is:

between t N~ and p N~ , (i = 1, 2, ..., p; j = 1, 2, ..., q).

G 0 = (w1  01 , w2  02 , …, wq  0q ) = (w1, w2, …, wq)

Here, σ  [0, 1] represents the environmental or resolution coefficient and it is used to adjust the difference of the relation coefficient. Generally, we set σ = 0.5. Step-2. Grey correlation coefficient matrix 

 12  12 . .

 12

G- between every alternative and INEURS is presented as follows:



between

every alternative and INERS is formulated as given below.

 11    21    = . .    p1 

Weighted neutrosophic grey correlation coefficient matrix

...  1q    ...  2q   ... .  ... .     ...  pq 

and correlation coefficient between INERS and INERS is:

 w1 11 w2 12 ...   w2 12 ...  w1 21  G- = . . ... . . ...     w1 p1 w2 12 ...  and similarly, weighted

wq  1q    wq  2q   .   .    wq  pq  correlation coefficient between

INEURS and INEURS is presented as follows:  G0 = (w1  01 , w2  02 , …, wq  0q ) = (w1, w2, …, wq)

Partha Pratim Dey, Surapati Pramanik, and Bibhas C. Giri, Neutrosophic soft multi-attribute decision making based on grey relational projection method

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Neutrosophic Sets and Systems, Vol. 11, 2016

Table 2. Tabular form of criterion decision matrix

Step-4. Calculation of the weighted grey correlation of alternative gi onto the INERS can be obtained as:  i

q  j1

( w j )

 2 ij



Cos

q  j1

(( w j ij )  w j )

q  j1

w 2j

q  j1

( w 2j  ij )

beautiful

(0.6,0.3, 0.8)

(0.7, 0.2, 0.6)

(0.8, 0.3, 0.4)

(0.7, 0.5, 0.6)

(0.8, 0.6, 0.7)

q  j1

( w j ij ) 2 

)

(( w j ij )  w j )

q  j1

(Gi,

 0

(10) q  j1

w 2j

Similarly, the weighted grey correlation of alternative gi onto the INEURS can be obtained as follows: Pri

=||

q  j1

( w j )

 2 ij



Cos

q  j1

(( w j ij )  w j )

q  j1

G 0

(Gi, =

( w j ) 

q  j1

(( w j ij )  w j )

q  j1

 2 ij

)

q  j1

2 j

q  j1

2 j

(11)

2 j

Step-5. Calculation of the neutrosophic relative relational degree The ranking order of all alternatives can be obtained according to the value of the neutrosophic relative relational degree. We calculate the neutrosophic relative relational degree by using the following equation Pr  Ci =  i  , i = 1, 2, …, p. (12) Pri  Pri Rank the alternatives according to the values of Ci, i = 1, 2, …, p in descending order and choose the alternative with biggest Ci. 4 A numerical example We consider the decision making problem for selecting the most suitable house for Mr. X [21]. Let Mr. X desires to select the most suitable house out of p houses on the basis of q parameters. Also let, the rating of or performance value of the house gi, i = 1, 2, ..., p with respect to parameter aj, j = 1, 2, …, q is represented by t N~ = ( TG (f ) (g i ) , I G (f ) (g i ) , FG (f ) (g i ) ) such that for a fixed i, t N~ denotes neutrosophic soft set of all the q objects. Let, ij A = {beautiful, cheap, in good repairing, moderate, wooden} be the set of choice parameters. The criterion decision matrix (see Table 2) is presented as follows: ij

(0.5, 0.2, 0.6) (0.6, 0.3, 0.7) (0.8, 0.5, 0.1) (0.6, 0.8, 0.7) (0.5, 0.6, 0.8)

in good repairing (0.7, 0.3, .4)

moderate

wooden

(0.8, 0.5, 0.6)

(0.6, 0.7, 0.2)

(0.7, 0.5, .6)

(0.6, 0.8, 0.3)

(0.8, 0.1, 0.8)

(0.3, 0.5, 0.6)

(0.7, 0.2, 0.1)

(0.7, 0.6, 0.8)

(0.8, 0.3, 0.6)

(0.8, 0.7,0.6)

(0.7, 0.8, 0.3)

(0.7, 0.2, 0.6) (0.8, 0.3, 0.8) (0.7, 0.2, 0.6)

The proposed procedure is presented in the following steps. Step 1. Calculation of the weights of the attribute

( w 2j  ij ) q  j1

cheap

Entropy value Ej (j = 1, 2, …, 5) of the j-th attribute can be obtained from the equation (3) as follows: E1 = 0.576, E2 = 0.556, E3 = 0.74, E4 = 0.564, E5 = 0.24. Then the corresponding normalized entropy weights are obtained as given below. w1 = 0.2155, w52 = 0.2076, w3 = 0.2763, w4 = 0.2111, w5 = 0.0895, where  w j = 1. j1 Step 2. Calculation of INERS and INEURS The INERS ( PN~ ) and INEURS ( PN~ ) of the decision matrix are shown as follows: PN~ = < (0.8, 0.2, 0.4); (0.8, 0.2, 0.1); (0.8, 0.3, 0.4); (0.8, 0.2, 0.1); (0.8, 0.1, 0.2) > PN~ = < (0.6, 0. 6, 0.8); (0.5, 0.8, 0.8); (0.3, 0.7, 0.8); (0.6, 0.8, 0.6); (0.6, 0.7, 0.8) > Step 3. Determine the grey relational coefficient of each alternative from INERS and INEURS The grey relational coefficient of each alternative from INERS can be determined as follows: 0.532  0.571 0.532 1.000 0.532 0.799 0.500 0.665 0.470 0.615    ij = 1.000 0.799 0.5 00 1.000 0.615    0.380 0.532 0.615 0.532  0.615 0.571 0.380 0.615 0.500 0.615    Similarly, the neutrosophic grey relational coefficient of each alternative from INEURS can be obtained as given below.

Partha Pratim Dey, Surapati Pramanik, and Bibhas C. Giri, Neutrosophic soft multi-attribute decision making based on grey relational projection method

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Neutrosophic Sets and Systems, Vol.11, 2016

1.000 0.555   ij = 0.483  0.714 1.000 

0.516

0.405

0.650

0.555

0.516

0.788

0.384

0.714

0.405

0.880

0.650

0.555

0.880

0.555

0.714

0.599  0.516  0.516   0.599  0.516 

Step 4. Calculation of the weighted grey correlation projection Calculation of the weighted grey correlation projection of alternative gi onto the INERS and INEURS can be obtained from the equations (10) and (11) respectively as follows: Pr1 = 0.1538, Pr2 = 0.1353, Pr3 = 0.1686, Pr4 = 0.1172, Pr5 = 0.117; Pr1 = 0.1333, Pr2 = 0.1283, Pr3 = 0.1157, Pr4 = 0.1502, Pr5 = 0.1627. Step 5. Calculate the grey relative relational degree We compute the grey relative relational degree by using equation (12) as follows: C1 = 0.5357, C2 = 0.5133, C3 = 0.5930, C4 = 0.4188, C5 = 0.4183. Step 6. The ranking order of the houses can be obtained according to the value of grey relative relational degree. It is observed that C3 > C1 > C2 > C4 > C5 and so the highest value of grey relative relational degree is C3. Therefore, the house g3 is the best alternative for Mr. X. Note: We now compare our proposed method with the method discussed by Maji [21]. Maji [21] first constructed the comparison matrix and then computed the score Si of gi,  i. The preferable alternative is selected based on the maximum score of Si. The ranking order of the houses is given by g5 > g3 > g4 > g1 > g2. In the present paper, a neutrosophic soft MADM problem through grey correlation projection method is proposed with unknown weights information. The ranking of alternatives are determined by the relative closeness to INERS which combines grey relational projection values from INERS and INEURS to each alternative. The ranking order of the houses is presented as g3 > g1 > g2 > g4 > g5. However, if he rejects the house h3 for any reason, his next preference will be g1. 5 Conclusion In this paper, we have presented a new approach for solving neutrosophic soft MADM problem based on GRP method with unknown weight information of the choice parameters. The proposed approach is a hybrid model of neutrosophic soft sets and GRP method where the choice parameters are represented in terms of single valued neutrosophic information. The weights of the parameters are determined by using information entropy method. In

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Partha Pratim Dey, Surapati Pramanik, and Bibhas C. Giri, Neutrosophic soft multi-attribute decision making based on grey relational projection method